Transport coefficients of leptons in superconducting neutron star cores

I consider the thermal conductivity and shear viscosity of leptons (electrons and muons) in the nucleon NS cores where protons are in the superconducting state. I restrict the consideration to the case of not too high temperatures $T\lesssim 0.35T_{\mathrm{c}p}$, where $T_{\mathrm{c}p}$ is the critical temperature of the proton pairing. In this case, lepton collisions with protons can be neglected. Charged lepton collision frequencies are mainly determined by the transverse plasmon exchange and are mediated by the character of the transverse plasma screening. In our previous works [Shternin \& Yakovlev, Phys. Rev. D {\bf 75} 103004 (2007); {\bf 78} 063006 (2008)] the superconducting proton contribution to the transverse screening was considered in the Pippard limit $\Delta \ll \hbar q v_{\mathrm{F}p}$, where $\Delta$ is the proton pairing gap, $v_{\mathrm{F}p}$ is the proton Fermi velocity, and $\hbar q$ is the typical transferred momentum in collisions. However, for large critical temperatures (large $\Delta$) and relatively small densities (small $q$) the Pippard limit may become invalid. In the present study I show that this is indeed the case and that the older calculations severely underestimated the screening in a certain range of the parameters appropriate to the neutron star cores. As a consequence, the kinetic coefficients at $T\ll T_{\mathrm{c}p}$ are found to be smaller than in previous calculations.